Resummed mean-field inference for strongly coupled data

TL;DR

This paper introduces a resummed mean-field method for inferring parameters of Ising and Potts models from noisy data, offering improved stability and accuracy over traditional methods, with low computational complexity and broad applicability.

Contribution

The authors develop a novel resummed mean-field approximation that enhances inference stability and accuracy for strongly coupled models, outperforming existing methods even without regularization.

Findings

Method outperforms standard mean-field inference techniques.

Inference remains stable across the entire phase diagram.

Algorithm is computationally efficient, requiring only matrix operations.

Abstract

We present a resummed mean-field approximation for inferring the parameters of an Ising or a Potts model from empirical, noisy, one- and two-point correlation functions. Based on a resummation of a class of diagrams of the small correlation expansion of the log-likelihood, the method outperforms standard mean-field inference methods, even when they are regularized. The inference is stable with respect to sampling noise, contrarily to previous works based either on the small correlation expansion, on the Bethe free energy, or on the mean-field and Gaussian models. Because it is mostly analytic, its complexity is still very low, requiring an iterative algorithm to solve for $N$ auxiliary variables, that resorts only to matrix inversions and multiplications. We test our algorithm on the Sherrington-Kirkpatrick model submitted to a random external field and large random couplings, and demonstrate that even without regularization, the inference is stable across the whole phase diagram. In addition, the calculation leads to a consistent estimation of the entropy of the data and allows us to sample form the inferred distribution to obtain artificial data that are consistent with the empirical distribution.